On subtractive varieties,V: congruence modularity and the commutators

In a congruence modular subtractive variety there are both the commutator of ideals and the commutator of congruences. We prove that, if I^' is the smallest congruence having an ideal I as a congruence class, then [I,J] = 0 /[I^', J^']. The general identity [0/ !,0 / #] = 0/[!,#] for !, # congruences, does not always hold; we give several conditions equivalent to this identity and sufficient conditions for it to hold. In the meantime, we get some other characterizations of the commutator of ideals. We also deal with the equational definability of principal commutators in a subtractive variety and with the extension property of the commutator from ideals of a subalgebra to the commutator of ideals of the whole algebra.     

The structure of commutative semigroups with the ideal extension property

In this paper, we will characterize commutative semigroups which have the ideal extension property (IEP). This characterization describes the multiplicative structure of commutative semigroups with IEP. Establishing this characterization was motivated not only by an interest in IEP itself, but also by the fact that in the category of commutative semigroups, the congruence extension property (CEP) implies IEP. A few preliminary results which hold in the general (non-commutative) case are discussed below. Following these initial observations, all semigroups considered are commutative.     

Many-sorted algebras in congruence modular varieties

We present several basic results on many-sorted algebras, most of them only valid in congruence modular varieties. We describe a connection between the properties of many-sorted varieties and those of varieties of one sort and give some results on functional completeness, the commutator and Abelian algebras.Presented by H. P. Gumm.     

Commuting Quasihomogeneous Toeplitz Operators on the Harmonic Bergman Space

In this paper, we obtain a symmetry number for the commutator of quasihomogeneous Toeplitz operators on the harmonic Bergman space. Then we use it to characterize the commuting Toeplitz operators with quasihomogeneous symbols. Also, we show that a Toeplitz operator with an analytic or co-analytic monomial symbol commutes with another Toeplitz operator only in the trivial case.     

On subtractive varieties IV: Definability of principal ideals

This paper deals with notions of (equational) definability of principal ideals in subtractive varieties. These notions are first characterized in several different ways. The strongest notion (EDPI) is then further investigated. We introduce the variety of MINI algebras (a generalization of Hilbert algebras) and we show that they are a paradigm for subtractive EDPI varieties. Finally we deal with principal ideal operations, and in particular with the cases of meet and join of principal ideals being equationally definable. Received November 7, 1996; accepted in final form December 17, 1997.     

Implicative Ideals and Fuzzy Implicative Ideals of a Distributive Implication Groupoid

We introduce the notions of implicative ideals and fuzzy implicative ideals of a distributive implication groupoid. Some properties of these ideals will be investigated. In particular, the necessary and sufficient conditions for an ideal (fuzzy ideal) to be an implicative ideal (fuzzy implicative ideal) is given. By using the concept of level sets, we will characterize the fuzzy implicative ideals of a distributive implication groupoid. Finally, an extension property for fuzzy implicative ideals is given.     

Resolutions of subsets of finite sets of points in projective space

Abstract

Eisenbud et al. proved a number of results regarding Gröbner bases and initial ideals of those ideals J in the free associative algebra K ?X ₁,…, X _n ? which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one weak notion of generic initial ideals in K ?X ₁,…, X _n ?, and show that generic initial ideals of ideals containing the anti-commutator ideal, or the commutator ideal, are finitely generated.     

The commutator algebra of a nilpotent matrix and an application to the theory of commutative Artinian algebras

We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian complete intersections have the strong Lefschetz property.     

Noncommutative Generalizations of Theorems of Cohen and Kaplansky

This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.     

BCH-代数的Ω-模糊点理想

引入BCH-代数的Ω-模糊点理想的概念,给出了它的几个实例,研究了它们的一些性质.讨论了Ω-模糊理想与Ω-模糊点理想的关系,获得了Ω-模糊点理想的几个等价描述,研究了Ω-模糊点理想的同态像与同态原像的性质,给出了BCH-代数的Ω-模糊点理想与BCH-代数之积代数的Ω-模糊点理想的关系, 讨论了模糊点理想与Ω-模糊点理想之间的相互构造.     

On subtractive varieties III: From ideals to congruences

As a sequel to [2] and [15] we investigate ideal properties focusing on subtractive varieties. Here we probe the relations between congruences and ideals in subtractive varieties, in order to give some means to recover the congruence structure from the ideal structure. To do so we consider mainly two operators from the ideal lattice to the congruence lattice of a given algebra and we classify subtractive varieties according to various properties of these operators. In the last section several examples are discussed in details. Received May 23, 1996; accepted in final form November 25, 1996.     

Torsion Abelian afi-Groups

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.     

Torsion Abelian RAI-Groups

A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. An Abelian group is called an RAI-group if there exists a ring on it in which every ideal is absolute. The problem of describing RAI-groups was formulated by L. Fuchs (Problem 93). In this paper, absolute ideals of torsion Abelian groups and torsion Abelian RAI-groups are described.     

On the Ideals of Secant Varieties of Segre Varieties

We establish basic techniques for determining the ideals of secant varieties of Segre varieties.We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.     

关于二次Putnam-Fuglede定理(Ⅰ)

设Ｎ为正规算子,若Ｎ与交换子ＮＸ－ＸＮ可交换,则Ｎ必与Ｘ可交换,称此结论为二次Ｐｕｔｎａｍ－Ｆｕｇｌｅｄｅ定理．本文给出了在幂零算子扰动下及在一些非正常算子时的二次ＰＦ定理     

Generic lattice ideals

A concept of genericity is introduced for lattice ideals (and hence for ideals defining toric varieties) which ensures nicely structured homological behavior. For a generic lattice ideal we construct its minimal free resolution and we show that it is induced from the Scarf resolution of any reverse lexicographic initial ideal.

     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Segments and Hilbert schemes of points

Using results obtained from a study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to a degreverse term order (as, for example, the degrevlex order) in the Hilbert scheme of points in Pⁿ. In this context, we look into the properties of several types of “segment” ideals that we define and compare. This study also leads us to focus on the connections between the shape of generators of Borel ideals and the related Hilbert polynomial, thus providing an algorithm for computing all saturated Borel ideals with a given Hilbert polynomial.